| 关于非线性反应扩散方程全局吸引子的整体与局部几何拓扑结构的研究 |
| Alternative Title | The Global and Local Geometric and Topological Structure of Global Attractor for Nonlinear Reaction-Diffusion Equations
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| 岳高成 |
| Thesis Advisor | 钟承奎
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| 2010-05-24
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 博士
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| Keyword | 全局吸引子的分解
复形
同调群
中心流形
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| Abstract | 在这篇博士学位论文中, 我们主要研究了下列非线性反应扩散方程全局吸引子的整体与局部几何拓扑结构, 得到了对全局吸引子几何拓扑结构的新的描述.
全文共分五章:
第一章, 介绍无穷维动力系统的理论和应用的背景, 全局吸引子问题的发展及研究进展情况, 总结全局吸引子存在性、维数估计和惯性流形的已有的理论和方法以及动力系统几何拓扑理论方面已有的成果.
第二章, 给出了本文用到的一些基础知识.
第三章, 主要研究了半线性反应扩散方程I 当外力项 时, 是相空间中的稠密子集(正则值集合), 全局吸引子的整体几何拓扑结构,得到了对全局吸引子的新的刻画, 也就是说, 方程I 的全局吸引子是平衡点的Lipschitz 连续的不稳定流形的并, 在一定程度上克服了方程I 在惯性流形不存在时对全局吸引子的几何结构的刻画所带来的困难, 这能很好地反映半线性反应扩散方程I 的全局吸引子的整体几何拓扑结构.
第四章, 主要研究了在第三章中得到的全局吸引子的代数和拓扑结构, 通过充分考虑全局吸引子自身所具有的性质, 受文献[111] 中关于建立Witten复形理论的启发, 在我们所得到的半线性反应扩散方程I 的全局吸引子A 上建立了Witten 同调群. 并证明了A 具有CW 复形结构, 得到了Witten 同调群、胞腔同调以及奇异同调群之间的同构关系, 这给出了奇异同调群的一种有效的计算方法. 最后, 结合全局吸引子的Morse过滤结构和相对同调群理论, 我们给出了全局吸引子的相对同调群的刻画, 得到了Morse 等式.
第五章, 主要研究了一类具有任意阶多项式增长的非线性反应扩散方程II的全局吸引子的局部几何拓扑结构, 即如果方程II 的线性化方程的谱和虚轴相交时, 我们所考虑的非线性反应扩散方程II 将出现中心流形, 我们得到了中心流形定理. |
| Other Abstract | In this doctoral dissertation, we study the global and local geometric and topological structure for global attractor of the following nonlinear reaction-diffusion equations,This thesis consists of five chapters.
In Chapter 1,we introduce the background of the theory and its applications of infinite dimensional dynamical systems, and the evolution of global attractor,and then, the method and theory of the existence of global attractor,dimensional estimate, inertial manifold,and the basic theory of geometry and topology of dynamical systems are listed in this chapter.
In Chapter 2,some preliminary results and definitions that we will used in this thesis are presented.
In Chapter 3,we mainly study the global geometric and topological structure of the corresponding global attractor for semilinear reaction-diffusion equations I when the forcing term g belongs to;where is an open and dense subset(regular value set) of the phase space;that is,the global attractor for equations I can be decomposed into the union of Lipschitz continuous manifold of equilibrium points.In some sense,we overcome some difficulties to describe the geometric structure of global attractor when the inertial manifold does not exist for equations I.In this way our decomposition for global attrator of equations I gives a good description on the geometric and topological structure of global attractor for semilinear reaction-diffusion equations I.
In Chapter4,we mainly study the algebraic and topological structure of global attractor obtained in Chapter3.We are motivated by the reference[111], in which Witten gave a beautiful method to establish Witten complex.Based on the theory of Witten complex,we establish Witten homology group on the global attractor A of the reaction-diffusion equations I when;and prove the global attractor A possesses the structure of CW
complex.We derive the isomorphism relation between any two of Witten homology group,cell homology group and singular homology group, which give an efficient tool to calculate the singular homology group. Last, by using the structure of Morse filtration and the theory of relative homology group,we give a description of relative homology group to the global attractor A and obtain the corresponding Morse's equation.
In Chapter5,we mainly study the local geometric and topological structure of global attractor for nonlinear reaction-diffusion equations II with a polynomial growth nonlinearity of arbitrary order,that is,if the spectrum o... |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/225357
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
岳高成. 关于非线性反应扩散方程全局吸引子的整体与局部几何拓扑结构的研究[D]. 兰州. 兰州大学,2010.
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